Exercise · ZW-CHARACTERISTIC-OF-A-RING-009 · ★★★ Advanced
If \(R\) has identity and characteristic \(n>0\), what is \(n\cdot a\) for any \(a\in R\)?
Choose one option.
Show answer
- C. \(0\)
Explanation
We have \(n\cdot a=(n\cdot1)a=0\cdot a=0\), so the characteristic annihilates the whole ring, not just 1.
Review the concept: Characteristic of a ring
The characteristic of a ring \(R\) with identity \(1\) is the least positive integer \(n\) such that adding the identity to itself \(n\) times gives zero; if none exists, it is 0.
\[\operatorname{char}(R)=\min\{\,n>0 : n\cdot1=0\,\}\]
More practice problems: characteristic of a ring
- ZW-CHARACTERISTIC-OF-A-RING-001 What is the characteristic of ℤ_12?
- ZW-CHARACTERISTIC-OF-A-RING-002 What is the characteristic of ℤ_5×ℤ_7?
- ZW-CHARACTERISTIC-OF-A-RING-003 What is the characteristic of ℤ_4×ℤ_6?
- ZW-CHARACTERISTIC-OF-A-RING-004 What is the characteristic of the ring ℤ[x] of polynomials with integer coefficients?
- ZW-CHARACTERISTIC-OF-A-RING-005 What is the characteristic of the field with 4 elements mathbbF_4=mathbbF_2[x]/(x^2+x+1)?
- ZW-CHARACTERISTIC-OF-A-RING-006 In ℤ_5, which element equals 23·1?
- ZW-CHARACTERISTIC-OF-A-RING-007 Which of these numbers can be the characteristic of a finite integral domain?
- ZW-CHARACTERISTIC-OF-A-RING-008 What is the characteristic of the matrix ring M_2(ℤ_3)?
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